Blocks adjustment—reduction of bias and variance of detrended fluctuation analysis using Monte Carlo simulation

The length of minimal and maximal blocks equally distant on log-log scale versus fluctuation function considerably influences bias and variance of DFA. Through a number of extensive Monte Carlo simulations and different fractional Brownian motion/fractional Gaussian noise generators, we found the pair of minimal and maximal blocks that minimizes the sum of mean-squared error of estimated Hurst exponents for the series of length . Sensitivity of DFA to sort-range correlations was examined using ARFIMA(p,d,q) generator. Due to the bias of the estimator for anti-persistent processes, we narrowed down the range of Hurst exponent to      

Measuring multifractality of stock price fluctuation using multifractal detrended fluctuation analysis

Analyzing the Shanghai stock price index daily returns using MF-DFA method, it is found that there are two different types of sources for multifractality in time series, namely, fat-tailed probability distributions and non-linear temporal correlations. Based on that, a sliding window of 240 frequency data in 5 trading days was used to study stock price index fluctuation. It is found that when the stock price index fluctuates sharply, a strong variability is clearly characterized by the generalized Hurst exponents h(q). Therefore, two measures, and σ, based on generalized Hurst exponents were proposed to compare financial risks before and after Price Limits and Reform of Non-tradable Shares. The empirical results verify the validity of the measures, and this has led to a better understanding of complex stock markets.     

Scaling and long-range dependence in option pricing I: Pricing European option with transaction costs under the fractional Black–Scholes model

This paper deals with the problem of discrete time option pricing by the fractional Black–Scholes model with transaction costs. By a mean self-financing delta-hedging argument in a discrete time setting, a European call option pricing formula is obtained. The minimal price of an option under transaction costs is obtained as timestep , which can be used as the actual price of an option. In fact, is an adjustment to the volatility in the Black–Scholes formula by using the modified volatility to replace the volatility σ, where is the Hurst exponent, and k is a proportional transaction cost parameter. In addition, we also show that timestep and long-range dependence have a significant impact on option pricing.     

Detrending moving average algorithm: A closed-form approximation of the scaling law

The Hurst exponent H of long range correlated series can be estimated by means of the detrending moving average (DMA) method. The computational tool, on which the algorithm is based, is the generalized variance , with being the average over the moving window n and N the dimension of the stochastic series y(i). The ability to yield H relies on the property of to vary as n^2H over a wide range of scales [E. Alessio, A. Carbone, G. Castelli, V. Frappietro, Eur. J. Phys. B 27 (2002) 197]. Here, we give a closed form proof that is equivalent to C_Hn^2H and provide an explicit expression for C_H. We furthermore compare the values of C_H with those obtained by applying the DMA algorithm to artificial self-similar signals.     

The local Hurst exponent of the financial time series in the vicinity of crashes on the Polish stock exchange market

We investigate the local fractal properties of the financial time series based on the whole history evolution (1991–2007) of the Warsaw Stock Exchange Index (WIG), connected with the largest developing financial market in Europe. Calculating the so-called local time-dependent Hurst exponent for the WIG time series we find the dependence between the behavior of the local fractal properties of the WIG time series and the crashes’ appearance on the financial market. We formulate the necessary conditions based on the behavior which have to be satisfied if the rupture or crash point is expected soon. As a result we show that the signal to sell or the signal to buy on the stock exchange market can be translated into evolution pattern. We also find a relation between the rate of the drop and the total correction the WIG index gains after the crash. The current situation on the market, particularly related to the recent Fed intervention in September ’07, is also discussed.     

Fluctuation scaling of quotation activities in the foreign exchange market

We study the scaling behavior of quotation activities for various currency pairs in the foreign exchange market. The components’ centrality is estimated from multiple time series and visualized as a currency pair network. The power-law relationship between a mean of quotation activity and its standard deviation for each currency pair is found. The scaling exponent α and the ratio between common and specific fluctuations η increase with the length of the observation time window . The result means that although for , the market dynamics are governed by specific processes, and at a longer time scale the common information flow becomes more important. We point out that quotation activities are not independently Poissonian for , and temporally or mutually correlated activities of quotations can happen even at this time scale. A stochastic model for the foreign exchange market based on a bipartite graph representation is proposed.     

Kolmogorov dispersion for turbulence in porous media: A conjecture

We will utilise the self-avoiding walk (SAW) mapping of the vortex line conformations in turbulence to get the Kolmogorov scale dependence of energy dispersion from SAW statistics, and the knowledge of the effects of disordered fractal geometries on the SAW statistics. These will give us the Kolmogorov energy dispersion exponent value for turbulence in porous media in terms of the size exponent for polymers in the same. We argue that the exponent value will be somewhat less than for turbulence in porous media.     

Multifractal analysis of the fracture surfaces of foamed polypropylene/polyethylene blends

The two-dimensional multifractal detrended fluctuation analysis is applied to reveal the multifractal properties of the fracture surfaces of foamed polypropylene/polyethylene (PP/PE) blends at different temperatures. Nice power-law scaling relationship between the detrended fluctuation function F_q and the scale s is observed for different orders q and the scaling exponent h(q) is found to be a nonlinear function of q, confirming the presence of multifractality in the fracture surfaces. The multifractal spectra f(α) are obtained numerically through Legendre transform. The shape of the multifractal spectrum of singularities can be well captured by the width of spectrum and the difference of dimension . With the increase of the PE content, the fracture surface becomes more irregular and complex, as is manifested by the facts that increases and decreases from positive to negative. A qualitative interpretation is provided based on the foaming process.     

Random walks and diameter of finite scale-free networks

Dynamical scalings for the end-to-end distance R_ee and the number of distinct visited nodes N_v of random walks (RWs) on finite scale-free networks (SFNs) are studied numerically. 〈R_ee〉 shows the dynamical scaling behavior , where is the average minimum distance between all possible pairs of nodes in the network, N is the number of nodes, γ is the degree exponent of the SFN and t is the step number of RWs. Especially, in the limit t→∞ satisfies the relation , where d is the diameter of network with for γ≥3 or for γ<3. Based on the scaling relation 〈R_ee〉, we also find that the scaling behavior of the diameter of networks can be measured very efficiently by using RWs.     

Entanglement degree measure and a criterion for sudden death as a phase transition in bipartite systems

We propose a method to compute the entanglement degree of bipartite systems having dimension 2 × 2 and demonstrate that the partial transposition of density matrix, the Peres criterion, arise as a consequence of our method. Differently from other existing measures of entanglement, the one presented here makes possible the derivation of a criterion to verify if an arbitrary bipartite entanglement will suffers sudden death (SD) based only on the initial-state parameters. Our method also makes possible to characterize the SD as a dynamical quantum phase transition, with order parameter , having a universal critical exponent −1/2.     

The dynamics of traded value revisited

We conclude from an analysis of high resolution NYSE data that the distribution of the traded value f_i (or volume) has a finite variance σ_i for the very large majority of stocks i, and the distribution itself is non-universal across stocks. The Hurst exponent of the same time series displays a crossover from weakly to strongly correlated behavior around the time scale of 1 day. The persistence in the strongly correlated regime increases with the average trading activity 〈f_i〉 as , which is another sign of non-universal behavior. The existence of such liquidity dependent correlations is consistent with the empirical observation that σ_i∝〈f_i〉^α, where α is a non-trivial, time scale dependent exponent.     

Boltzmann distribution and market temperature

We show that the minute fluctuations of S&P 500 and NASDAQ 100 indices show Boltzmann statistics over a wide range of positive as well as negative returns, thus allowing us to define a market temperature for either sign. With increasing time the sharp Boltzmann peak broadens into a Gaussian whose volatility σ measured in is related to the temperature T by . Plots over the years 1990-2006 show that the arrival of the 2000 crash was preceded by an increase in market temperature, suggesting that this increase can be used as a warning signal for crashes.     

Analytical computation of amplification of coupling in relativistic equations with Yukawa potential

The approximate analytic solutions to the Klein-Gordon and Dirac equations with the Yukawa potential were derived by using the quasilinearization method (QLM). The accurate analytic expressions for the ground state energies and wave functions were presented. These high-precision approximate analytic representations are obtained by first casting the proper relativistic equation into a nonlinear Riccati form and then solving that nonlinear equation in the first QLM iteration. The choice of zero iteration is based on general features of the exact solutions near the origin and infinity. To estimate the accuracy of the QLM solutions, the exact numerical solutions were found, as well. The analytical QLM solutions are found to be extremely accurate for a small exponent parameter w of the Yukawa potential. The reasonable accuracy is kept for the medium values of w. When w approaches the critical values, the precision of the QLM results falls down markedly. However, the approximate analytic QLM solution to the Dirac equation corresponding to the maximum relativistic effect turned out to be very accurate even for w close to the exact critical , whereas the QLM calculations yield . This effect of “amplification” in compare with the Schrödinger equation critical parameter was investigated earlier [S. De Leo, P. Rotelli, Phys. Rev. D 69 (2004) 034006]. In this work, it was found that the “amplification” for the Klein-Gordon equation became all the more evident. The exact numerical value is , whereas the QLM approximation yields .     

Sufficient and necessary condition of separability for generalized Werner states

In a celebrated paper [Optics Communications 179, 447, 2000], A.O. Pittenger and M.H. Rubin presented for the first time a sufficient and necessary condition of separability for the generalized Werner states. Inspired by their ideas, we generalized their method to a more general case. We obtain a sufficient and necessary condition for the separability of a specific class of N d-dimensional system (qudits) states, namely special generalized Werner state (SGWS): , where is an entangled pure state of N qudits system and α_i satisfies two restrictions: (i) ; (ii) Matrix , where , is a density matrix. Our condition gives quite a simple and efficiently computable way to judge whether a given SGWS is separable or not and previously known separable conditions are shown to be special cases of our approach.     

Anomalous thermostat and intraband discrete breathers

We investigate the dynamics of a macroscopic system which consists of an anharmonic subsystem embedded in an arbitrary harmonic lattice, including quenched disorder. The coupling between both parts is bilinear. Elimination of the harmonic degrees of freedom leads to a nonlinear Langevin equation with memory kernels and noise term for the anharmonic coordinates . For zero temperature, i.e. for , we prove that the support of the Fourier transform of and of the time averaged velocity-velocity correlation functions of the anharmonic system cannot overlap. As a consequence, the asymptotic solutions can be constant, periodic, quasiperiodic or almost periodic, and possibly weakly chaotic. For a sinusoidal trajectory with frequency we find that the energy E_T transferred to the harmonic system up to time T is proportional to T^α. If equals one of the phonon frequencies ω_ν, it is α=2. We prove that there is a zero measure set L such that for in its full measure complement R?L, it is α=0, i.e. there is no energy dissipation. Under certain conditions L contains a subset L^′ such that for the dissipation rate is nonzero and may be subdissipative (0≤α<1) or superdissipative (1<α≤2), compared to ordinary dissipation (α=1). Consequently, the harmonic bath does act as an anomalous thermostat, in variance with the common belief that elimination of a macroscopically large number of degrees of freedom always generates dissipation, forcing convergence to equilibrium. Intraband discrete breathers are such solutions which do not relax. We prove for arbitrary anharmonicity and small but finite coupling that intraband discrete breathers with frequency exist for all in a Cantor set C(k) of finite Lebesgue measure. This is achieved by estimating the contribution of small denominators appearing for , related to . For the small denominators do not lead to divergencies such that is a smooth and bounded function in t.     

Multifractal Detrended Cross-Correlation Analysis of sunspot numbers and river flow fluctuations

We use the Detrended Cross-Correlation Analysis (DCCA) to investigate the influence of sun activity represented by sunspot numbers on one of the climate indicators, specifically rivers, represented by river flow fluctuation for Daugava, Holston, Nolichucky and French Broad rivers. The Multifractal Detrended Cross-Correlation Analysis (MF-DXA) shows that there exist some crossovers in the cross-correlation fluctuation function versus time scale of the river flow and sunspot series. One of these crossovers corresponds to the well-known cycle of solar activity demonstrating a universal property of the mentioned rivers. The scaling exponent given by DCCA for original series at intermediate time scale, , is λ=1.17±0.04 which is almost similar for all underlying rivers at 1σ confidence interval showing the second universal behavior of river runoffs. To remove the sinusoidal trends embedded in data sets, we apply the Singular Value Decomposition (SVD) method. Our results show that there exists a long-range cross-correlation between the sunspot numbers and the underlying streamflow records. The magnitude of the scaling exponent and the corresponding cross-correlation exponent are λ∈(0.76,0.85) and γ_×∈(0.30,0.48), respectively. Different values for scaling and cross-correlation exponents may be related to local and external factors such as topography, drainage network morphology, human activity and so on. Multifractal cross-correlation analysis demonstrates that all underlying fluctuations have almost weak multifractal nature which is also a universal property for data series. In addition the empirical relation between scaling exponent derived by DCCA and Detrended Fluctuation Analysis (DFA), is confirmed.